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Mirrors > Home > MPE Home > Th. List > itgeq2dv | Structured version Visualization version GIF version |
Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.) |
Ref | Expression |
---|---|
itgeq2dv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
itgeq2dv | ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itgeq2dv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
2 | 1 | ralrimiva 3103 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
3 | itgeq2 24942 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥) |
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